Introduction to coalgebra. Towards mathematics of states and observation (Q2828459)

Coalgebras have developed into an important tool for investigating state-based systems; the most quoted paper in this area was for quite some time probably \textit{J. J. M. M. Rutten}'s [Theor. Comput. Sci. 249, No. 1, 3--80 (2000; Zbl 0951.68038)] which introduced coalgebras to a wider audience. Well, now we have Bart Jacobs' new book, the evolution of which could be traced and was discussed by the community for some time.NEWLINENEWLINEA coalgebra \((K, c)\) for a functor \(F\) on a category \(K\) is just a \(C\)-morphism \(c: K\to F(K)\); this notion is dual to that of an algebra \((A, a)\) which is a morphism \(a: F(A) \to A\). Many important examples from theoretical computer science fit into this pattern. Jacobs starts gently with his presentation, giving examples along the way, introducing categories, functors and many of the tools required for later on working on (co)algebras. The author does not expect the reader to be familiar with categories at all (only the frequently required, somewhat mystical mathematical maturity is assumed), he introduces the tools as the topics unfold. The pace in the first three chapters is leisurely, the emphasis being on polynomial functors, final coalgebras and bisimulations. Chapter 4 opens the gallery of more advanced topics, working in particular on the manifold and colorful connections between coalgebras, algebras and logics. Clearly, finality plays a very important role, the question of the existence of a final coalgebra is discussed in some detail, and the expressivity of coalgebras (bisimilarity, logical and behavioral equivalence among them) is investigated from various vantage points. The important role of polynomial functors is stressed, analytical functors, which are closely related, are discussed; both families are characterized in terms of preservation properties. So far, monads have not been discussed in detail, this is done in Chapter 5, where monads, comonads, Kleisli categories are introduced; a nice application to bialgebras and operational semantics is given and exhibited in considerable detail. The last chapter deals with the important topic of invariants and assertions, giving among others the connection between invariants and limits of coalgebras on the one hand, and to modal logics on the other. The final section on coalgebraic class specifications may be of particular interest to the object-oriented community.NEWLINENEWLINEThe book is very carefully written, its many examples are discussed usually in great detail, expressing patiently the salient features to be exhibited. This applies to the usually fairly detailed proofs as well. Another treasure trove is the collection of most instructive exercises, of which there are plenty. Occasionally, this reviewer would have preferred a slightly different approach, e.g., it is sometimes helpful in teaching to introduce monads through Kleisli triples, and the discussion of the neighborhood monad (which in the disguise of effectivity functions serve the interpretation of game logics well) could have been more extensive. But these are just very minor points.NEWLINENEWLINEMost of the development is based on functors on the category of sets, although the author occasionally also refers to general categories. This reviewer would have appreciated greater attention to the development around the Giry monad for probabilistic coalgebras (it is mentioned in some places), on the other hand the tools from measure theory for a careful discussion of stochastic coalgebras are not always readily available to the intended audience. Here, \textit{P. Panangaden}'s [Labelled Markov processes. London: Imperial College Press (2009; Zbl 1190.60001)], or the reviewer's [Special topics in mathematics for computer scientists. Sets, categories, topologies and measures. Cham: Springer (2015; Zbl 1334.68002)], might be a suitable complement. Nevertheless, instead of referring to coalgebras, a hint at set-based coalgebras in the title would have made life easier for those of us who work on the sometimes subtle differences between the properties of set-based and other coalgebras.NEWLINENEWLINEThis admirable book sets the standard for the work on set-based coalgebras for many years to come. It can be used for reference, and it is equally usable as a textbook.